This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
5.2 Uniform Circular motion5.3 Dynamic of Uniform Circular Motion
• A) Your mass• B) Your weight• C) Your apparent weight• D) A and B• E) B and C
Applications of Newton’s Laws Involving Friction
Example 5-5: Two boxes and a pulley
Two boxes are connected by a cord running over a pulley. The coefficient of kinetic friction between box A and the table is 0.20. We ignore the mass of the cord and pulley and any friction in the pulley, which means we can assume that a force applied to one end of the cord will have the same magnitude at the other end. We wish to find the acceleration, a, of the system, which will have the same magnitude for both boxes assuming the cord doesn’t stretch. As box B moves down, box A moves to the right.
Uniform Circular Motion—Kinematics
Uniform circular motion: motion in a circle of constant radius at constant speed
Instantaneous velocity is always tangent to the circle.
• How far does it go?– Angular displacement, to linear
motion, s.
– Here r is the radius of the circle in meters, and s is the distance traveled in meters (or arc length).
is the angular displacement in radians
since s/r is unitless, radians are not a physical unit, and do not need to balance like most units.
y
x
i
f
s
r
Relationship between angular and linear motion
How fast does it go?– Angular velocity, , to linear velocity,
v
– Direction of v is tangent to the circle– Units :
v m/sec must be in rad/s
y
x
i
f
r
v
v
Questio
n• Two objects are sitting on a horizontal
table that is undergoing uniform circular motion. Assuming the objects don’t slip, which of the following statements is true?
• A) Objects 1 and 2 have the same linear velocity, v, and the same angular velocity, .
• B) Objects 1 and 2 have the same linear velocity, v, and the different angular velocities, .
• C) Objects 1 and 2 have different linear velocities, v, and the same angular velocity, .
• D) Objects 1 and 2 have different linear velocities, v, and the different angular velocities, .
1
2
Questio
n• Two objects are sitting on a horizontal
table that is undergoing uniform circular motion. Assuming the objects don’t slip, which of the following statements is true?
• A) Objects 1 and 2 have the same linear velocity, v.
• B) Object 1 has a faster linear velocity than object 2.
• C) Object 1 has a slower linear velocity than object 2.
1
2
Period and frequency
Linear vs. Circular motion:
Linear motion Angular motion
Position Angle
Displacement Angular displacement
Average velocity Average angular velocity
Instantaneous velocity
Instantaneous angular velocity
Dynamics of Uniform Circular Motion
• Direction: towards the center of the circle
Velocity can be constant in magnitude, and we still have acceleration because the direction changes.
Radial Acceleration: magnitude
As t gets small, v becomes an arc length
since
and
ar = ac
Newton’s second law
• Whenever we have circular motion, we have acceleration directed towards the center of the motion.
• Whenever we have circular motion, there must be a force towards the center of the circle causing the circular motion.
Dynamics of Uniform Circular Motion
There is no centrifugal force pointing outward; what happens is that the natural tendency of the object to move in a straight line must be overcome.
If the centripetal force vanishes, the object flies off at a tangent to the circle.
Dynamics of Uniform Circular Motion
Example 5-11: Force on revolving ball (horizontal).
Estimate the force a person must exert on a string attached to a 0.150-kg ball to make the ball revolve in a horizontal circle of radius 0.600 m. The ball makes 2.00 revolutions per second. Ignore the string’s mass.
Dynamics of Uniform Circular Motion
Example 5-13: Conical pendulum.
A small ball of mass m, suspended by a cord of length l, revolves in a circle of radius r = l sin θ, where θ is the angle the string makes with the vertical. (a) In what direction is the acceleration of the ball, and what causes the acceleration? (b) Calculate the speed and period (time required for one revolution) of the ball in terms of l, θ, g, and m.
Highway Curves: Banked and Unbanked
When a car goes around a curve, there must be a net force toward the center of the circle of which the curve is an arc. If the road is flat, that force is supplied by friction.
Highway Curves: Banked and Unbanked
If the frictional force is insufficient, the car will tend to move more nearly in a straight line, as the skid marks show.
Highway Curves: Banked and Unbanked
As long as the tires do not slip, the friction is static. If the tires do start to slip, the friction is kinetic, which is bad in two ways:
1. The kinetic frictional force is smaller than the static.
2. The static frictional force can point toward the center of the circle, but the kinetic frictional force opposes the direction of motion, making it very difficult to regain control of the car and continue around the curve.
Highway Curves: Banked and Unbanked
Example 5-14: Skidding on a curve.
A 1000-kg car rounds a curve on a flat road of radius 50 m at a speed of 15 m/s (54 km/h). Will the car follow the curve, or will it skid? Assume: (a) the pavement is dry and the coefficient of static friction is μs = 0.60; (b) the pavement is icy and μs = 0.25.
Highway Curves: Banked and Unbanked
Banking the curve can help keep cars from skidding. In fact, for every banked curve, there is one speed at which the entire centripetal force is supplied by the
horizontal component of the normal force, and no friction is required. This occurs when:
Non-uniform Circular Motion
If an object is moving in a circular path but at varying speeds, it must have a tangential component to its acceleration as well as the radial one.
Non-uniform Circular Motion
This concept can be used for an object moving along any curved path, as any small segment of the path will be approximately circular.